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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclscls00 | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.) |
Ref | Expression |
---|---|
ntrcls.o | ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) |
ntrcls.d | ⊢ 𝐷 = (𝑂‘𝐵) |
ntrcls.r | ⊢ (𝜑 → 𝐼𝐷𝐾) |
Ref | Expression |
---|---|
ntrclscls00 | ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrcls.o | . . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
2 | ntrcls.d | . . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵) | |
3 | ntrcls.r | . . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
4 | 1, 2, 3 | ntrclsfv1 44045 | . . . . 5 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾) |
5 | 4 | fveq1d 6909 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐾‘∅)) |
6 | 2, 3 | ntrclsbex 44024 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ V) |
7 | 1, 2, 3 | ntrclsiex 44043 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
8 | eqid 2735 | . . . . 5 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼) | |
9 | 0elpw 5362 | . . . . . 6 ⊢ ∅ ∈ 𝒫 𝐵 | |
10 | 9 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∅ ∈ 𝒫 𝐵) |
11 | eqid 2735 | . . . . 5 ⊢ ((𝐷‘𝐼)‘∅) = ((𝐷‘𝐼)‘∅) | |
12 | 1, 2, 6, 7, 8, 10, 11 | dssmapfv3d 44009 | . . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅)))) |
13 | 5, 12 | eqtr3d 2777 | . . 3 ⊢ (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅)))) |
14 | dif0 4384 | . . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵 | |
15 | 14 | fveq2i 6910 | . . . . . 6 ⊢ (𝐼‘(𝐵 ∖ ∅)) = (𝐼‘𝐵) |
16 | id 22 | . . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘𝐵) = 𝐵) | |
17 | 15, 16 | eqtrid 2787 | . . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵) |
18 | 17 | difeq2d 4136 | . . . 4 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵 ∖ 𝐵)) |
19 | difid 4382 | . . . 4 ⊢ (𝐵 ∖ 𝐵) = ∅ | |
20 | 18, 19 | eqtrdi 2791 | . . 3 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅) |
21 | 13, 20 | sylan9eq 2795 | . 2 ⊢ ((𝜑 ∧ (𝐼‘𝐵) = 𝐵) → (𝐾‘∅) = ∅) |
22 | pwidg 4625 | . . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵) | |
23 | 6, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵) |
24 | 1, 2, 3, 23 | ntrclsfv 44049 | . . 3 ⊢ (𝜑 → (𝐼‘𝐵) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵)))) |
25 | 19 | fveq2i 6910 | . . . . . 6 ⊢ (𝐾‘(𝐵 ∖ 𝐵)) = (𝐾‘∅) |
26 | id 22 | . . . . . 6 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅) | |
27 | 25, 26 | eqtrid 2787 | . . . . 5 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘(𝐵 ∖ 𝐵)) = ∅) |
28 | 27 | difeq2d 4136 | . . . 4 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = (𝐵 ∖ ∅)) |
29 | 28, 14 | eqtrdi 2791 | . . 3 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = 𝐵) |
30 | 24, 29 | sylan9eq 2795 | . 2 ⊢ ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼‘𝐵) = 𝐵) |
31 | 21, 30 | impbida 801 | 1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ∅c0 4339 𝒫 cpw 4605 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 |
This theorem is referenced by: (None) |
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